Optimized aerodynamic profile for a turbine vane, in particular for a nozzle of a turbine

ABSTRACT

When cold and in the non-coated state, the aerodynamic profile is substantially identical to a nominal profile determined by the Cartesian coordinates X,Y, Zadim given in Table 1, in which the coordinate Zadim is the quotient D/H where D is the distance of the point under consideration from a first reference plane P0 situated at the base of the nominal profile, and H is the height of said profile measured from the first reference plane to a second reference plane P1. The measurements D and H are taken radially relative to the axis of the turbine, while the X coordinate is measured in the axial direction of the turbine.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of priority to U.S. Patent Application No. 62/504,555, filed on May 11, 2017, the entirety of which is incorporated by reference herein.

INTRODUCTION

The present disclosure relates to an aerodynamic profile for a turbine vane.

In particular, the disclosure relates to the aerodynamic profile of a nozzle vane forming a portion of the stator of a gas turbine, and more particularly of a low pressure turbine of the type used in a turbojet of an aircraft.

The disclosure relates in particular to a nozzle vane of a turbine having a plurality of stages, preferably five stages.

Such a profile should enable the turbine to provide the desired efficiency, and in order to do that it must be such that the flow of air around the profile is sound over the working spectrum of the turbine, i.e. substantially such that it does not give rise to turbulence, which is harmful for overall efficiency.

It must be capable of being installed properly in the environment of the engine, and in particular, for a nozzle vane forming a portion of the stator of the turbine, it must be capable of being fastened easily to the sectors of the foot platform (the zone of the vane that is furthest from the axis of rotation of the turbine) and of the head platform (the zone that is closest to said axis). Furthermore, this part must present a profile enabling it to be manufactured in a reliable and cost-effective manner by available manufacturing methods, such as casting, forging, machining, additive fabrication, or else weaving, without this list being limitative.

Furthermore, the profile of the vane must enable it to withstand the mechanical stresses, to which the vane is subjected, by enabling those stresses to be spread over the entire vane in such a manner as to avoid premature wear thereof. This spreading must apply both in static mode and in dynamic mode.

An object of the disclosure is to propose an aerodynamic profile for a turbine vane that is optimized, and capable of satisfying those objectives.

This object is achieved by the fact that when cold and in a non-coated state, said profile is substantially identical to a nominal profile determined by the Cartesian coordinates X,Y,Zadim given in Table 1, in which the coordinate Zadim is the quotient D/H where D is the distance of the point P under consideration from a reference X,Y plane situated at the base of the nominal profile, and H is the height of said profile measured from said reference plane that is the intersection of the stacking axis of the vane and the axisymmetric surface of the hub, out to a second reference plane that is the intersection of said stacking axis with the axisymmetric surface of the casing, the measurements D and H being taken radially relative to the axis of the turbine, while the coordinate X is measured in the axial direction of the turbine.

This profile has been determined as a result of numerous tests and simulations. It is defined cold, i.e. at an ambient temperature of 20° C. That is a reference temperature at which the profile is geometrically determined. The above-mentioned objectives of aerodynamics and mechanical optimization are naturally valid for the conditions of use of this aerodynamic profile, i.e. when hot, at a temperature that is stabilized when the engine of which the turbine forms a part is being used under cruising conditions.

Furthermore, the aerodynamic profile of the disclosure is defined in the non-coated state. Since turbine vanes are subjected to high temperature gradients, it is common practice for them to be provided with a coating having thermal properties enabling them more easily to withstand such temperature variations. The profile is determined prior to installing such a coating.

It is stated above that the profile of the disclosure is “substantially identical” to the nominal profile. This means that the profile may depart very slightly from said nominal profile.

The aerodynamic profile is thus preferably defined within an envelope of ±1 millimeter (mm) in a direction normal to the surface of the nominal profile.

It is also preferable for the X,Y coordinates of the aerodynamic profile to lie within a range of ±5% relative to the X,Y coordinates of the nominal profile.

In addition, the aerodynamic profile may be positioned rotated within ±10°, particularly within ±5°, more particularly within ±3°, with respect to the radial axis of the vane, along which the coordinate Zadim is measured.

These variations take account in particular to at least one of manufacturing tolerances of the profile, the setting of the profile to adapt to the environment when in operation, and possible deformation occurring when mounting the vane and integrating the same in the nozzle.

With a nozzle vane, i.e. a portion of the stator of a turbine, the profile serves to orient the fluid correctly for the turbine blades that are situated downstream, and that belong in particular to the rotary wheel of the same stage of the turbine.

BRIEF DESCRIPTION OF THE DRAWINGS

The disclosure can be better understood and its advantages appear better on reading the following detailed description of an embodiment given by way of non-limiting example. The description refers to the accompanying drawings, in which:

FIG. 1 is a fragmentary axial diagrammatic section view of a turbine including a vane of profile that corresponds to that of the present disclosure;

FIG. 2 shows a vane of the turbine.

DETAILED DESCRIPTION

The turbine shown in axial section in FIG. 1 comprises five stages referenced respectively S1 to S5. In the direction DF going from upstream to downstream, each stage comprises a nozzle that forms a portion of the stator of the turbine and that has a plurality of radially-oriented vanes, and a rotary wheel that forms a portion of the rotor of the turbine and that likewise includes a plurality of radially-oriented blades.

In FIG. 1, the vanes of the nozzles of stages S1 to S5 are given references N1 to N5 (these vanes are shown in dashed lines, for ease of representation), whereas the blades of rotary wheels of stages S1 to S5 are given respective references W1 to W5.

In known manner, the vanes of the nozzles are fastened at both ends to stationary structure portions, while the blades of the rotary wheels are fastened to a rotary disk via their roots that are formed at their radially-inner ends closer to the axis of the rotation A of the turbine. The disks forming parts of the wheels W1 to W5 are constrained to rotate together.

The disclosure relates a vane of one of the nozzles of the turbine that, as mentioned above, preferably comprises five stages as shown, without this number being limiting.

As best seen in FIG. 2, via its foot 10, which is its end remote from the axis of rotation A, the vane is fastened to the outer ring of the turbine, which ring is stationary in rotation. Via its head 14, formed by its end closer to the axis of rotation A, the vane is fastened to an inner shroud of the turbine. By convention, the aerodynamic profile 18 of the vane is the entire portion of said vane that extends radially outwards from its head 14 to its foot 10 without incorporating the fastenings respectively to the outer ring and to the inner shroud.

In FIG. 2, there can be seen a frame of reference comprising Cartesian coordinates X,Y,Z. The radial direction Z is the height direction of the vane, which extends radially from its head to its foot. This direction Z is perpendicular to the axial direction X which is the direction of the axis of rotation A of the turbine. The direction Y is perpendicular to the X,Z plane and is therefore tangential to the direction of rotation of the turbine.

The nominal profile from which the aerodynamic profile of the disclosure is determined is defined in following Table 1 of coordinates, in which the coordinate Zadim, measured along the axis Z is non-dimensional, whereas the dimensions X and Y, respectively measured along the axes X and Y, are expressed in millimeters.

TABLE 1 X Y Zadim −9.5839 1.645034 −0.15001 −9.59598 1.589313 −0.15001 −9.60652 1.511004 −0.15001 −9.61338 1.396335 −0.15001 −9.60916 1.238254 −0.15001 −9.57983 1.030729 −0.15001 −9.5049 0.773888 −0.15001 −9.36183 0.477387 −0.15001 −9.13411 0.159274 −0.15001 −8.82452 −0.16962 −0.15001 −8.4383 −0.50389 −0.15001 −7.9754 −0.83437 −0.15001 −7.43501 −1.14891 −0.15001 −6.81645 −1.4305 −0.15001 −6.12208 −1.65982 −0.15001 −5.35873 −1.81567 −0.15001 −4.53862 −1.87793 −0.15001 −3.67949 −1.82872 −0.15001 −2.80164 −1.65516 −0.15001 −1.9286 −1.35529 −0.15001 −1.07914 −0.93538 −0.15001 −0.26761 −0.40739 −0.15001 0.497644 0.212281 −0.15001 1.212444 0.907124 −0.15001 1.875688 1.660956 −0.15001 2.489151 2.458009 −0.15001 3.056167 3.283933 −0.15001 3.582129 4.125161 −0.15001 4.070232 4.971248 −0.15001 4.521753 5.813917 −0.15001 4.94068 6.644155 −0.15001 5.331095 7.453775 −0.15001 5.694305 8.236638 −0.15001 6.030805 8.987528 −0.15001 6.342829 9.70094 −0.15001 6.631382 10.37249 −0.15001 6.896441 10.99879 −0.15001 7.138259 11.57691 −0.15001 7.357444 12.10441 −0.15001 7.553943 12.57973 −0.15001 7.727732 13.00205 −0.15001 7.879223 13.37111 −0.15001 8.008862 13.68743 −0.15001 8.117302 13.95227 −0.15001 8.205454 14.16772 −0.15001 8.27312 14.33738 −0.15001 8.270783 14.47239 −0.15001 8.226619 14.55616 −0.15001 8.184018 14.59771 −0.15001 8.145323 14.62163 −0.15001 8.114813 14.63447 −0.15001 8.072877 14.64476 −0.15001 8.00426 14.64628 −0.15001 7.907352 14.61685 −0.15001 7.829071 14.51035 −0.15001 7.752344 14.35935 −0.15001 7.658885 14.17321 −0.15001 7.546969 13.95094 −0.15001 7.415913 13.69169 −0.15001 7.265288 13.39509 −0.15001 7.09499 13.06119 −0.15001 6.902191 12.69211 −0.15001 6.678154 12.29385 −0.15001 6.413552 11.87331 −0.15001 6.103207 11.43646 −0.15001 5.745373 10.98811 −0.15001 5.340396 10.5323 −0.15001 4.889814 10.07286 −0.15001 4.395703 9.613708 −0.15001 3.86139 9.158171 −0.15001 3.290874 8.709701 −0.15001 2.689107 8.270893 −0.15001 2.061575 7.844907 −0.15001 1.413965 7.433854 −0.15001 0.752777 7.039262 −0.15001 0.084802 6.662034 −0.15001 −0.58566 6.306489 −0.15001 −1.25587 5.97872 −0.15001 −1.92456 5.688206 −0.15001 −2.58804 5.440664 −0.15001 −3.24139 5.237669 −0.15001 −3.87984 5.072355 −0.15001 −4.49958 4.931448 −0.15001 −5.09928 4.794919 −0.15001 −5.67943 4.644155 −0.15001 −6.24007 4.460894 −0.15001 −6.77627 4.236557 −0.15001 −7.28237 3.975456 −0.15001 −7.74922 3.686103 −0.15001 −8.16917 3.379806 −0.15001 −8.53694 3.06946 −0.15001 −8.84986 2.767144 −0.15001 −9.10516 2.481738 −0.15001 −9.29901 2.22313 −0.15001 −9.43046 2.006613 −0.15001 −9.50694 1.850771 −0.15001 −9.5473 1.752576 −0.15001 −9.44092 2.198932 −0.075 −9.46805 2.151808 −0.075 −9.49987 2.08036 −0.075 −9.53627 1.964925 −0.075 −9.56321 1.79494 −0.075 −9.55607 1.564893 −0.075 −9.48481 1.283283 −0.075 −9.33114 0.970038 −0.075 −9.0911 0.644384 −0.075 −8.78244 0.303388 −0.075 −8.40714 −0.05181 −0.075 −7.96257 −0.41441 −0.075 −7.44366 −0.77055 −0.075 −6.8466 −1.10232 −0.075 −6.17069 −1.38878 −0.075 −5.41984 −1.60679 −0.075 −4.60374 −1.73297 −0.075 −3.73839 −1.74581 −0.075 −2.84394 −1.62946 −0.075 −1.94525 −1.37634 −0.075 −1.06459 −0.99053 −0.075 −0.21961 −0.48468 −0.075 0.579094 0.12252 −0.075 1.326266 0.811851 −0.075 2.020646 1.564836 −0.075 2.664164 2.364097 −0.075 3.260436 3.194166 −0.075 3.814421 4.041113 −0.075 4.329551 4.893886 −0.075 4.807592 5.743642 −0.075 5.2524 6.581166 −0.075 5.66753 7.398274 −0.075 6.054442 8.188573 −0.075 6.413748 8.946647 −0.075 6.74731 9.666991 −0.075 7.056003 10.34515 −0.075 7.339836 10.97763 −0.075 7.598978 11.56148 −0.075 7.833863 12.09422 −0.075 8.044455 12.5743 −0.075 8.230769 13.00083 −0.075 8.39317 13.37357 −0.075 8.532134 13.69305 −0.075 8.648359 13.96053 −0.075 8.742834 14.17815 −0.075 8.815592 14.34944 −0.075 8.82003 14.48699 −0.075 8.778251 14.57394 −0.075 8.736056 14.61744 −0.075 8.697184 14.64261 −0.075 8.66491 14.6565 −0.075 8.620416 14.6671 −0.075 8.548132 14.66632 −0.075 8.450075 14.62857 −0.075 8.368454 14.51425 −0.075 8.283007 14.35559 −0.075 8.178644 14.16008 −0.075 8.053772 13.92659 −0.075 7.907698 13.65417 −0.075 7.740005 13.34242 −0.075 7.550577 12.99141 −0.075 7.336941 12.60314 −0.075 7.09254 12.18274 −0.075 6.811548 11.73554 −0.075 6.489949 11.26677 −0.075 6.125511 10.78151 −0.075 5.719145 10.28334 −0.075 5.272398 9.775835 −0.075 4.786819 9.263053 −0.075 4.264689 8.74895 −0.075 3.709297 8.237033 −0.075 3.124032 7.731063 −0.075 2.512913 7.234749 −0.075 1.880085 6.751475 −0.075 1.230005 6.284604 −0.075 0.566908 5.838048 −0.075 −0.106 5.417298 −0.075 −0.78688 5.030658 −0.075 −1.47409 4.688656 −0.075 −2.16404 4.401174 −0.075 −2.85083 4.17489 −0.075 −3.5267 4.007105 −0.075 −4.18393 3.887296 −0.075 −4.81705 3.800177 −0.075 −5.42238 3.727296 −0.075 −5.99754 3.652345 −0.075 −6.53912 3.565624 −0.075 −7.04344 3.462137 −0.075 −7.50591 3.341352 −0.075 −7.9225 3.2058 −0.075 −8.28948 3.059321 −0.075 −8.60321 2.905208 −0.075 −8.86156 2.749173 −0.075 −9.06368 2.599004 −0.075 −9.2118 2.466212 −0.075 −9.31214 2.361104 −0.075 −9.3765 2.285137 −0.075 −9.22394 2.908108 0 −9.26228 2.86814 0 −9.30676 2.801029 0 −9.34969 2.684469 0 −9.36753 2.512301 0 −9.34316 2.287609 0 −9.26762 2.01569 0 −9.13507 1.70191 0 −8.94082 1.351693 0 −8.68034 0.971396 0 −8.34894 0.568973 0 −7.94186 0.154647 0 −7.45481 −0.25883 0 −6.88398 −0.65502 0 −6.22725 −1.01341 0 −5.48594 −1.30959 0 −4.66675 −1.51579 0 −3.78445 −1.60442 0 −2.8604 −1.55291 0 −1.92388 −1.35042 0 −1.00183 −1.00067 0 −0.11527 −0.51912 0 0.723517 0.072146 0 1.509004 0.750641 0 2.240443 1.495529 0 2.92036 2.288056 0 3.552855 3.111989 0 4.142465 3.953474 0 4.692619 4.801266 0 5.205532 5.646103 0 5.684875 6.47875 0 6.133467 7.291286 0 6.552697 8.077231 0 6.943235 8.831038 0 7.306492 9.54732 0 7.643074 10.22166 0 7.952942 10.85054 0 8.236142 11.431 0 8.492902 11.96063 0 8.723158 12.43788 0 8.926936 12.86186 0 9.104567 13.23234 0 9.256548 13.54987 0 9.383657 13.81571 0 9.486967 14.03197 0 9.56683 14.20207 0 9.580384 14.34175 0 9.542809 14.43221 0 9.501813 14.47853 0 9.46297 14.50583 0 9.428693 14.5216 0 9.381033 14.53341 0 9.303163 14.53096 0 9.202399 14.4833 0 9.116082 14.36181 0 9.019387 14.19777 0 8.901022 13.99576 0 8.759525 13.7545 0 8.594179 13.47302 0 8.404603 13.1509 0 8.190687 12.78828 0 7.950624 12.38695 0 7.680095 11.95094 0 7.374958 11.48499 0 7.032993 10.99343 0 6.653259 10.48061 0 6.236594 9.95014 0 5.784504 9.405592 0 5.298463 8.850977 0 4.78038 8.290326 0 4.232551 7.727682 0 3.657589 7.167097 0 3.058017 6.613001 0 2.43649 6.069983 0 1.79583 5.542739 0 1.138914 5.036189 0 0.467122 4.557712 0 −0.21843 4.116171 0 −0.91679 3.722561 0 −1.6258 3.389072 0 −2.34083 3.127967 0 −3.05233 2.944062 0 −3.74881 2.835936 0 −4.41881 2.794821 0 −5.05297 2.805532 0 −5.6456 2.850652 0 −6.19418 2.913175 0 −6.69744 2.979516 0 −7.15505 3.039384 0 −7.56664 3.085707 0 −7.93174 3.115204 0 −8.24975 3.12616 0 −8.52036 3.119678 0 −8.74356 3.097554 0 −8.91977 3.06256 0 −9.05001 3.018395 0 −9.13739 2.972479 0 −9.21948 2.881951 0.1 −9.26016 2.842624 0.1 −9.30656 2.774701 0.1 −9.34834 2.654886 0.1 −9.36101 2.478481 0.1 −9.3292 2.249824 0.1 −9.24559 1.973867 0.1 −9.10545 1.655334 0.1 −8.9044 1.299003 0.1 −8.6379 0.910738 0.1 −8.30123 0.498062 0.1 −7.88969 0.070545 0.1 −7.39715 −0.35777 0.1 −6.8197 −0.7707 0.1 −6.15457 −1.14734 0.1 −5.40176 −1.46121 0.1 −4.56711 −1.6815 0.1 −3.66677 −1.77641 0.1 −2.72296 −1.71934 0.1 −1.76912 −1.49711 0.1 −0.83685 −1.1154 0.1 0.051548 −0.59301 0.1 0.88365 0.044603 0.1 1.655144 0.771967 0.1 2.366915 1.566169 0.1 3.023125 2.407213 0.1 3.629457 3.278066 0.1 4.190992 4.16493 0.1 4.712062 5.056295 0.1 5.196071 5.942437 0.1 5.646955 6.814279 0.1 6.067343 7.664202 0.1 6.459253 8.485512 0.1 6.823824 9.272487 0.1 7.162323 10.0199 0.1 7.47549 10.7233 0.1 7.76357 11.37904 0.1 8.02668 11.98413 0.1 8.264992 12.53621 0.1 8.478563 13.03364 0.1 8.667586 13.47549 0.1 8.832348 13.86155 0.1 8.973215 14.19244 0.1 9.090955 14.4695 0.1 9.186606 14.6949 0.1 9.260391 14.87213 0.1 9.272551 15.01502 0.1 9.231585 15.10636 0.1 9.187885 15.15199 0.1 9.146924 15.17796 0.1 9.109041 15.19272 0.1 9.05687 15.20155 0.1 8.97365 15.19003 0.1 8.874402 15.1228 0.1 8.791659 14.98496 0.1 8.694559 14.80419 0.1 8.57551 14.58175 0.1 8.433294 14.31601 0.1 8.267264 14.00588 0.1 8.077092 13.65084 0.1 7.862681 13.25102 0.1 7.623279 12.80757 0.1 7.356246 12.32334 0.1 7.061449 11.80046 0.1 6.738525 11.24179 0.1 6.38759 10.65045 0.1 6.008461 10.03028 0.1 5.601238 9.385431 0.1 5.166699 8.720094 0.1 4.705229 8.039124 0.1 4.217194 7.347816 0.1 3.703101 6.651767 0.1 3.162441 5.957777 0.1 2.595457 5.272596 0.1 2.002517 4.603418 0.1 1.382805 3.959242 0.1 0.735727 3.350027 0.1 0.059841 2.788514 0.1 −0.64497 2.289047 0.1 −1.37676 1.867998 0.1 −2.12893 1.541057 0.1 −2.88928 1.320169 0.1 −3.64121 1.209845 0.1 −4.36394 1.204043 0.1 −5.04167 1.28847 0.1 −5.66366 1.442167 0.1 −6.22421 1.641682 0.1 −6.7241 1.863902 0.1 −7.16724 2.089666 0.1 −7.55839 2.304636 0.1 −7.90201 2.498779 0.1 −8.20214 2.664611 0.1 −8.46154 2.797948 0.1 −8.68292 2.894674 0.1 −8.86781 2.951849 0.1 −9.01557 2.963489 0.1 −9.1193 2.940505 0.1 −9.19971 2.927162 0.2 −9.24259 2.888696 0.2 −9.29129 2.820582 0.2 −9.33405 2.698663 0.2 −9.34543 2.518906 0.2 −9.3109 2.286479 0.2 −9.22409 2.006004 0.2 −9.081 1.681883 0.2 −8.87737 1.318505 0.2 −8.60857 0.921502 0.2 −8.26961 0.498293 0.2 −7.85516 0.058717 0.2 −7.35984 −0.38429 0.2 −6.77864 −0.81387 0.2 −6.1078 −1.20828 0.2 −5.34649 −1.53984 0.2 −4.4998 −1.77537 0.2 −3.58395 −1.87973 0.2 −2.6231 −1.82308 0.2 −1.65328 −1.59091 0.2 −0.70914 −1.19035 0.2 0.186051 −0.64311 0.2 1.020412 0.022368 0.2 1.790748 0.778598 0.2 2.498915 1.601619 0.2 3.150679 2.470372 0.2 3.754823 3.36601 0.2 4.312457 4.277672 0.2 4.829186 5.193072 0.2 5.309165 6.102114 0.2 5.75627 6.995801 0.2 6.172755 7.866699 0.2 6.560966 8.707929 0.2 6.922274 9.513638 0.2 7.257653 10.27869 0.2 7.567829 10.99861 0.2 7.853166 11.66965 0.2 8.113759 12.28881 0.2 8.349738 12.85369 0.2 8.561231 13.36263 0.2 8.748377 13.8147 0.2 8.911337 14.20975 0.2 9.05054 14.54839 0.2 9.166987 14.8319 0.2 9.261613 15.06251 0.2 9.335357 15.24352 0.2 9.349184 15.38891 0.2 9.3077 15.48203 0.2 9.263029 15.52831 0.2 9.221061 15.55433 0.2 9.181165 15.56905 0.2 9.126346 15.5767 0.2 9.039962 15.56058 0.2 8.940959 15.48354 0.2 8.858474 15.33689 0.2 8.758704 15.14685 0.2 8.636132 14.91315 0.2 8.489801 14.63391 0.2 8.319103 14.30795 0.2 8.123763 13.93469 0.2 7.903625 13.51429 0.2 7.65898 13.04738 0.2 7.389626 12.53552 0.2 7.095478 11.98083 0.2 6.776539 11.38597 0.2 6.433136 10.75399 0.2 6.065831 10.0883 0.2 5.674633 9.393066 0.2 5.259517 8.672937 0.2 4.819667 7.933431 0.2 4.354064 7.180661 0.2 3.862126 6.420915 0.2 3.341622 5.662092 0.2 2.790527 4.91267 0.2 2.207093 4.181817 0.2 1.589239 3.480126 0.2 0.934272 2.82052 0.2 0.240652 2.217583 0.2 −0.49072 1.687094 0.2 −1.25564 1.245825 0.2 −2.04519 0.910113 0.2 −2.84446 0.692496 0.2 −3.63407 0.596729 0.2 −4.38947 0.616513 0.2 −5.09431 0.735952 0.2 −5.73485 0.933 0.2 −6.30482 1.182998 0.2 −6.80427 1.462311 0.2 −7.23744 1.750443 0.2 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8.777569 12.91039 0.3 9.002243 13.42276 0.3 9.201046 13.8779 0.3 9.374247 14.27562 0.3 9.522357 14.61651 0.3 9.646174 14.90192 0.3 9.746777 15.13411 0.3 9.82541 15.31627 0.3 9.845428 15.46316 0.3 9.806685 15.55914 0.3 9.762791 15.60748 0.3 9.720887 15.63499 0.3 9.680727 15.65078 0.3 9.625218 15.65942 0.3 9.537436 15.64375 0.3 9.437119 15.56526 0.3 9.351352 15.41759 0.3 9.245745 15.22708 0.3 9.116075 14.99275 0.3 8.961371 14.71271 0.3 8.781012 14.38574 0.3 8.574765 14.01125 0.3 8.342463 13.58939 0.3 8.084178 13.12093 0.3 7.800072 12.60728 0.3 7.490828 12.05019 0.3 7.156503 11.45234 0.3 6.79764 10.81673 0.3 6.414055 10.14725 0.3 6.005332 9.448427 0.3 5.571979 8.72471 0.3 5.113097 7.981818 0.3 4.62732 7.226243 0.3 4.114455 6.464227 0.3 3.572615 5.703668 0.3 2.999549 4.953468 0.3 2.393905 4.222795 0.3 1.754536 3.521753 0.3 1.079505 2.862784 0.3 0.368008 2.259717 0.3 −0.37852 1.72731 0.3 −1.1558 1.281227 0.3 −1.95543 0.936751 0.3 −2.76385 0.706435 0.3 −3.56339 0.596365 0.3 −4.33068 0.603951 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9.314325 0.4 6.772418 10.15694 0.4 7.084205 10.95653 0.4 7.372364 11.70859 0.4 7.637463 12.40929 0.4 7.879461 13.05563 0.4 8.098373 13.64528 0.4 8.29449 14.17647 0.4 8.467945 14.64828 0.4 8.618994 15.06053 0.4 8.748116 15.41385 0.4 8.856039 15.70966 0.4 8.9437 15.9503 0.4 9.012246 16.13905 0.4 9.026371 16.28842 0.4 8.982952 16.38378 0.4 8.936351 16.43064 0.4 8.892632 16.45637 0.4 8.851159 16.47021 0.4 8.794517 16.47571 0.4 8.706776 16.45403 0.4 8.611261 16.36735 0.4 8.533871 16.21261 0.4 8.438137 16.01378 0.4 8.320675 15.7692 0.4 8.180637 15.47688 0.4 8.017491 15.13555 0.4 7.831074 14.74458 0.4 7.621211 14.30411 0.4 7.388098 13.81485 0.4 7.131915 13.27817 0.4 6.853012 12.696 0.4 6.55125 12.07108 0.4 6.227082 11.40642 0.4 5.880271 10.70586 0.4 5.51001 9.973995 0.4 5.11647 9.215388 0.4 4.698504 8.435708 0.4 4.254143 7.641526 0.4 3.783071 6.838982 0.4 3.283429 6.035593 0.4 2.75237 5.240156 0.4 2.188199 4.461557 0.4 1.589116 3.709787 0.4 0.953733 2.995839 0.4 0.280387 2.332868 0.4 −0.43037 1.73468 0.4 −1.17562 1.21603 0.4 −1.94899 0.791771 0.4 −2.73971 0.475637 0.4 −3.53206 0.276142 0.4 −4.30609 0.19538 0.4 −5.0394 0.225289 0.4 −5.71739 0.348927 0.4 −6.32714 0.543039 0.4 −6.86468 0.782207 0.4 −7.33154 1.0432 0.4 −7.73318 1.306048 0.4 −8.07701 1.554831 0.4 −8.3712 1.777143 0.4 −8.62367 1.963284 0.4 −8.84104 2.105527 0.4 −9.02692 2.198465 0.4 −9.18212 2.236161 0.4 −9.29583 2.22556 0.4 −9.40706 2.336525 0.5 −9.45142 2.296477 0.5 −9.50165 2.225713 0.5 −9.54516 2.09871 0.5 −9.55428 1.912114 0.5 −9.51294 1.671848 0.5 −9.41398 1.383714 0.5 −9.25368 1.052897 0.5 −9.02775 0.684477 0.5 −8.73149 0.284874 0.5 −8.35991 −0.13753 0.5 −7.90769 −0.57162 0.5 −7.3696 −1.00275 0.5 −6.74108 −1.4118 0.5 −6.01951 −1.77419 0.5 −5.20628 −2.05934 0.5 −4.31193 −2.23182 0.5 −3.35551 −2.25691 0.5 −2.36664 −2.11052 0.5 −1.38411 −1.78877 0.5 −0.43553 −1.3068 0.5 0.461976 −0.69078 0.5 1.301482 0.03129 0.5 2.082737 0.833826 0.5 2.809447 1.694618 0.5 3.487011 2.594985 0.5 4.119537 3.52036 0.5 4.70994 4.458884 0.5 5.262874 5.398947 0.5 5.782329 6.330457 0.5 6.27017 7.245306 0.5 6.727425 8.136424 0.5 7.156322 8.996691 0.5 7.557755 9.820206 0.5 7.931624 10.6021 0.5 8.278314 11.33782 0.5 8.597965 12.02353 0.5 8.890381 12.65619 0.5 9.155427 13.23341 0.5 9.393125 13.75349 0.5 9.603527 14.21546 0.5 9.78687 14.61916 0.5 9.943682 14.96516 0.5 10.07478 15.25486 0.5 10.18131 15.49054 0.5 10.26466 15.67541 0.5 10.29259 15.82432 0.5 10.25772 15.92424 0.5 10.21522 15.9757 0.5 10.17359 16.00557 0.5 10.13299 16.02319 0.5 10.07632 16.03346 0.5 9.98604 16.01816 0.5 9.883854 15.93709 0.5 9.794426 15.78651 0.5 9.682587 15.59314 0.5 9.545351 15.35523 0.5 9.381726 15.07083 0.5 9.191102 14.73868 0.5 8.973289 14.35815 0.5 8.728139 13.92939 0.5 8.455927 13.45308 0.5 8.156947 12.93058 0.5 7.831733 12.3638 0.5 7.480357 11.75551 0.5 7.103506 11.10871 0.5 6.701315 10.42724 0.5 6.273279 9.71582 0.5 5.820025 8.978963 0.5 5.340865 8.222436 0.5 4.834902 7.452601 0.5 4.301202 6.676355 0.5 3.739559 5.90063 0.5 3.148159 5.134083 0.5 2.525511 4.385977 0.5 1.871363 3.66556 0.5 1.185472 2.98322 0.5 0.46742 2.351213 0.5 −0.28156 1.782509 0.5 −1.05784 1.290328 0.5 −1.85503 0.888012 0.5 −2.66314 0.587938 0.5 −3.46784 0.398654 0.5 −4.25224 0.322748 0.5 −4.99484 0.354587 0.5 −5.68058 0.478884 0.5 −6.29796 0.675033 0.5 −6.84175 0.91778 0.5 −7.31342 1.18353 0.5 −7.7187 1.451563 0.5 −8.06524 1.705485 0.5 −8.36166 1.932343 0.5 −8.61532 2.123327 0.5 −8.83453 2.268576 0.5 −9.02214 2.363688 0.5 −9.17904 2.402799 0.5 −9.29424 2.392905 0.5 −9.71888 1.346095 0.6 −9.7596 1.301656 0.6 −9.80446 1.22654 0.6 −9.84028 1.095812 0.6 −9.83859 0.907553 0.6 −9.78146 0.668252 0.6 −9.66035 0.385801 0.6 −9.47139 0.06673 0.6 −9.21056 −0.2826 0.6 −8.87378 −0.65459 0.6 −8.45696 −1.03973 0.6 −7.95612 −1.42571 0.6 −7.36781 −1.79673 0.6 −6.69007 −2.13274 0.6 −5.92371 −2.40864 0.6 −5.07437 −2.59439 0.6 −4.15875 −2.65813 0.6 −3.19556 −2.5713 0.6 −2.22083 −2.31834 0.6 −1.2648 −1.90171 0.6 −0.35023 −1.33993 0.6 0.510543 −0.65803 0.6 1.313567 0.118461 0.6 2.060163 0.966459 0.6 2.754727 1.865883 0.6 3.402622 2.799846 0.6 4.007263 3.755289 0.6 4.571537 4.720984 0.6 5.100226 5.68577 0.6 5.597099 6.64003 0.6 6.063484 7.576126 0.6 6.500485 8.48706 0.6 6.910411 9.365795 0.6 7.294031 10.20656 0.6 7.651181 11.00453 0.6 7.982345 11.75513 0.6 8.287748 12.45452 0.6 8.567141 13.09967 0.6 8.820319 13.68824 0.6 9.047358 14.21849 0.6 9.248316 14.68948 0.6 9.423422 15.10103 0.6 9.573186 15.45374 0.6 9.698397 15.74906 0.6 9.800138 15.98929 0.6 9.879704 16.17775 0.6 9.906487 16.32821 0.6 9.870617 16.42874 0.6 9.827553 16.48051 0.6 9.785424 16.51045 0.6 9.744172 16.52802 0.6 9.686676 16.53772 0.6 9.595455 16.52034 0.6 9.494591 16.43533 0.6 9.408113 16.28045 0.6 9.29969 16.08186 0.6 9.166677 15.83752 0.6 9.008128 15.54545 0.6 8.82347 15.20433 0.6 8.612539 14.81352 0.6 8.375196 14.37315 0.6 8.111722 13.88391 0.6 7.822386 13.34718 0.6 7.507722 12.76489 0.6 7.16777 12.13983 0.6 6.803121 11.47506 0.6 6.413968 10.77438 0.6 5.99967 10.04256 0.6 5.560702 9.284207 0.6 5.096775 8.50479 0.6 4.606123 7.711128 0.6 4.088619 6.909522 0.6 3.54396 6.10683 0.6 2.969991 5.311718 0.6 2.36521 4.533169 0.6 1.729735 3.779763 0.6 1.063205 3.061401 0.6 0.365095 2.389538 0.6 −0.36363 1.776121 0.6 −1.11997 1.233255 0.6 −1.89854 0.772954 0.6 −2.6914 0.407205 0.6 −3.48674 0.145247 0.6 −4.26915 −0.0088 0.6 −5.02194 −0.05723 0.6 −5.7257 −0.0116 0.6 −6.3706 0.109868 0.6 −6.94784 0.284859 0.6 −7.45607 0.490916 0.6 −7.89786 0.707892 0.6 −8.27949 0.917014 0.6 −8.6085 1.103266 0.6 −8.89095 1.256926 0.6 −9.13203 1.370009 0.6 −9.33581 1.433531 0.6 −9.49873 1.445376 0.6 −9.61251 1.416938 0.6 −10.0644 −0.15989 0.7 −10.0985 −0.21027 0.7 −10.1339 −0.2911 0.7 −10.1566 −0.42611 0.7 −10.1374 −0.61478 0.7 −10.0558 −0.84953 0.7 −9.90149 −1.11853 0.7 −9.67074 −1.41315 0.7 −9.36033 −1.72495 0.7 −8.96763 −2.04469 0.7 −8.4904 −2.3613 0.7 −7.92718 −2.66125 0.7 −7.27791 −2.92788 0.7 −6.54498 −3.14126 0.7 −5.73439 −3.27915 0.7 −4.85943 −3.31518 0.7 −3.93563 −3.22266 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0.7 6.740263 13.3319 0.7 6.429226 12.67954 0.7 6.095351 11.98555 0.7 5.738782 11.25371 0.7 5.358723 10.48875 0.7 4.955285 9.69543 0.7 4.528191 8.879116 0.7 4.075582 8.046432 0.7 3.596751 7.20385 0.7 3.091408 6.358033 0.7 2.557439 5.517209 0.7 1.99279 4.690249 0.7 1.397172 3.885592 0.7 0.770257 3.11247 0.7 0.111225 2.381586 0.7 −0.57955 1.704141 0.7 −1.30009 1.091683 0.7 −2.04598 0.555328 0.7 −2.81057 0.105851 0.7 −3.58416 −0.24768 0.7 −4.35392 −0.4996 0.7 −5.10438 −0.65114 0.7 −5.8196 −0.70986 0.7 −6.48503 −0.69071 0.7 −7.09291 −0.61291 0.7 −7.63781 −0.49742 0.7 −8.11968 −0.36441 0.7 −8.54138 −0.23208 0.7 −8.90743 −0.11587 0.7 −9.22205 −0.02789 0.7 −9.48852 0.019802 0.7 −9.7052 0.024414 0.7 −9.86729 −0.01168 0.7 −9.97184 −0.06869 0.7 −10.6896 −0.52801 0.8 −10.7108 −0.58624 0.8 −10.7347 −0.67317 0.8 −10.7535 −0.81089 0.8 −10.74 −1.00361 0.8 −10.6695 −1.24683 0.8 −10.519 −1.52453 0.8 −10.281 −1.82242 0.8 −9.95651 −2.13162 0.8 −9.54364 −2.4412 0.8 −9.0416 −2.73883 0.8 −8.45109 −3.01111 0.8 −7.77444 −3.24279 0.8 −7.01589 −3.41575 0.8 −6.18444 −3.50949 0.8 −5.29082 −3.50314 0.8 −4.3538 −3.37501 0.8 −3.39671 −3.11122 0.8 −2.44346 −2.7077 0.8 −1.51469 −2.17203 0.8 −0.62515 −1.51887 0.8 0.218036 −0.76791 0.8 1.012619 0.061405 0.8 1.759569 0.951069 0.8 2.462004 1.884722 0.8 3.123555 2.848122 0.8 3.745942 3.829971 0.8 4.331384 4.819712 0.8 4.883891 5.806674 0.8 5.405979 6.781835 0.8 5.898042 7.737889 0.8 6.361073 8.667723 0.8 6.796818 9.564358 0.8 7.205452 10.42213 0.8 7.586602 11.23614 0.8 7.940618 12.00174 0.8 8.267436 12.71505 0.8 8.566618 13.37305 0.8 8.837957 13.9733 0.8 9.081435 14.51404 0.8 9.297058 14.99432 0.8 9.48502 15.41398 0.8 9.645853 15.77362 0.8 9.780354 16.07473 0.8 9.88968 16.31967 0.8 9.975121 16.51184 0.8 10.01106 16.66482 0.8 9.981201 16.7702 0.8 9.940886 16.82661 0.8 9.899886 16.86042 0.8 9.858052 16.88154 0.8 9.798719 16.89474 0.8 9.70263 16.87897 0.8 9.598365 16.79005 0.8 9.505679 16.62971 0.8 9.389478 16.42396 0.8 9.246891 16.17084 0.8 9.076903 15.86826 0.8 8.878893 15.51487 0.8 8.652659 15.11 0.8 8.398093 14.65379 0.8 8.1155 14.14694 0.8 7.805206 13.59089 0.8 7.467816 12.98763 0.8 7.103516 12.33999 0.8 6.712919 11.65121 0.8 6.296494 10.92514 0.8 5.853886 10.1666 0.8 5.385469 9.380501 0.8 4.891091 8.572503 0.8 4.370322 7.748896 0.8 3.822584 6.916512 0.8 3.24707 6.082828 0.8 2.642377 5.2563 0.8 2.009414 4.444303 0.8 1.347201 3.656278 0.8 0.654953 2.902498 0.8 −0.06735 2.193869 0.8 −0.81928 1.542553 0.8 −1.59879 0.961379 0.8 −2.40073 0.462223 0.8 −3.21727 0.056253 0.8 −4.03674 −0.24897 0.8 −4.84443 −0.45115 0.8 −5.62422 −0.55506 0.8 −6.36124 −0.57576 0.8 −7.0447 −0.53291 0.8 −7.66887 −0.44912 0.8 −8.23223 −0.34666 0.8 −8.73614 −0.24548 0.8 −9.18287 −0.16154 0.8 −9.57411 −0.10802 0.8 −9.90976 −0.09911 0.8 −10.1846 −0.14442 0.8 −10.3906 −0.22998 0.8 −10.5289 −0.32966 0.8 −10.6138 −0.41786 0.8 −12.0981 −2.10799 0.9 −12.1121 −2.17258 0.9 −12.1258 −2.26662 0.9 −12.1311 −2.41194 0.9 −12.1054 −2.61507 0.9 −12.0158 −2.86603 0.9 −11.8359 −3.14516 0.9 −11.5561 −3.43236 0.9 −11.1821 −3.7203 0.9 −10.7148 −3.99699 0.9 −10.1547 −4.24929 0.9 −9.50486 −4.46457 0.9 −8.76986 −4.62823 0.9 −7.95643 −4.7249 0.9 −7.07478 −4.73566 0.9 −6.13897 −4.64178 0.9 −5.16737 −4.4254 0.9 −4.18246 −4.07488 0.9 −3.20688 −3.58845 0.9 −2.25964 −2.97388 0.9 −1.35497 −2.24424 0.9 −0.50114 −1.41664 0.9 0.298058 −0.50878 0.9 1.043637 0.461061 0.9 1.738972 1.476118 0.9 2.388473 2.521425 0.9 2.995285 3.584591 0.9 3.561632 4.654903 0.9 4.091709 5.721514 0.9 4.589406 6.774605 0.9 5.056049 7.806239 0.9 5.492756 8.809096 0.9 5.901842 9.775874 0.9 6.284306 10.70039 0.9 6.639995 11.5775 0.9 6.969556 12.4023 0.9 7.273315 13.17065 0.9 7.551043 13.87932 0.9 7.802682 14.52572 0.9 8.028336 15.10799 0.9 8.228105 15.62512 0.9 8.402199 16.07695 0.9 8.551138 16.46415 0.9 8.675682 16.78831 0.9 8.776912 17.052 0.9 8.855685 17.25896 0.9 8.891534 17.42063 0.9 8.861198 17.53115 0.9 8.818171 17.59044 0.9 8.774408 17.6253 0.9 8.729919 17.64628 0.9 8.667255 17.65764 0.9 8.56754 17.63424 0.9 8.46786 17.53119 0.9 8.37975 17.35809 0.9 8.270302 17.13564 0.9 8.136046 16.86197 0.9 7.976005 16.53488 0.9 7.789604 16.15291 0.9 7.576591 15.71539 0.9 7.336865 15.22245 0.9 7.070682 14.67484 0.9 6.778284 14.07411 0.9 6.460163 13.42238 0.9 6.116313 12.72266 0.9 5.747181 11.97837 0.9 5.353047 11.19356 0.9 4.933375 10.37322 0.9 4.48821 9.522531 0.9 4.016602 8.647671 0.9 3.516148 7.756092 0.9 2.983202 6.856592 0.9 2.412136 5.960105 0.9 1.796025 5.079786 0.9 1.128375 4.230458 0.9 0.405148 3.428059 0.9 −0.37441 2.689341 0.9 −1.21002 2.034797 0.9 −2.09489 1.483365 0.9 −3.01481 1.045673 0.9 −3.95035 0.721388 0.9 −4.88086 0.501052 0.9 −5.78765 0.363041 0.9 −6.65704 0.27852 0.9 −7.48048 0.217214 0.9 −8.25161 0.15174 0.9 −8.96455 0.059485 0.9 −9.61184 −0.07492 0.9 −10.1846 −0.25736 0.9 −10.6742 −0.48262 0.9 −11.0768 −0.73683 0.9 −11.3953 −1.00146 0.9 −11.6368 −1.2582 0.9 −11.8124 −1.4919 0.9 −11.9337 −1.69209 0.9 −12.0122 −1.85315 0.9 −12.0589 −1.97283 0.9 −13.6094 −3.68219 1 −13.594 −3.75853 1 −13.5722 −3.86233 1 −13.5367 −4.00683 1 −13.4711 −4.19512 1 −13.361 −4.42688 1 −13.1926 −4.69502 1 −12.9452 −4.99792 1 −12.5942 −5.32124 1 −12.122 −5.63602 1 −11.5283 −5.90446 1 −10.825 −6.09883 1 −10.0287 −6.20889 1 −9.15453 −6.23333 1 −8.21771 −6.16641 1 −7.23155 −6.00038 1 −6.21258 −5.71913 1 −5.18307 −5.3049 1 −4.16827 −4.74834 1 −3.18705 −4.0591 1 −2.25303 −3.25206 1 −1.37402 −2.34522 1 −0.5536 −1.35676 1 0.208402 −0.30448 1 0.914037 0.79529 1 1.567919 1.926914 1 2.17439 3.076818 1 2.736194 4.233581 1 3.257422 5.386083 1 3.743057 6.523723 1 4.195593 7.63775 1 4.616551 8.720392 1 5.008676 9.763972 1 5.373819 10.76177 1 5.712257 11.70821 1 6.024882 12.59813 1 6.31248 13.42708 1 6.575049 14.19155 1 6.812707 14.8888 1 7.025706 15.51683 1 7.214229 16.07456 1 7.378519 16.56182 1 7.519078 16.97935 1 7.636641 17.3289 1 7.732213 17.61321 1 7.806599 17.83601 1 7.843443 18.00831 1 7.814653 18.12621 1 7.769526 18.18973 1 7.723054 18.22676 1 7.674009 18.24889 1 7.605231 18.25858 1 7.498645 18.22293 1 7.403942 18.09822 1 7.317066 17.90452 1 7.20906 17.65593 1 7.076564 17.35016 1 6.918616 16.98474 1 6.734636 16.55807 1 6.524306 16.06943 1 6.287508 15.51896 1 6.024429 14.90752 1 5.735162 14.23684 1 5.419977 13.50934 1 5.078599 12.72837 1 4.711318 11.89769 1 4.318475 11.02165 1 3.899242 10.10567 1 3.450392 9.156866 1 2.964942 8.184864 1 2.431167 7.20293 1 1.832592 6.229425 1 1.151023 5.288595 1 0.371746 4.41008 1 −0.51319 3.62874 1 −1.5001 2.981355 1 −2.56651 2.491449 1 −3.6785 2.160906 1 −4.80348 1.97421 1 −5.91521 1.898493 1 −6.9982 1.881011 1 −8.04449 1.862709 1 −9.04345 1.778327 1 −9.97143 1.578484 1 −10.7989 1.256685 1 −11.5049 0.832385 1 −12.0858 0.342149 1 −12.5521 −0.18179 1 −12.9107 −0.70329 1 −13.1939 −1.23822 1 −13.3936 −1.74328 1 −13.522 −2.19482 1 −13.597 −2.58522 1 −13.6349 −2.91233 1 −13.6479 −3.17374 1 −13.6465 −3.37038 1 −13.6368 −3.51124 1 −14.326 −3.76941 1.075003 −14.2118 −4.22031 1.075003 −14.0665 −4.58722 1.075003 −13.9079 −4.88537 1.075003 −13.7373 −5.14138 1.075003 −13.5512 −5.37197 1.075003 −13.3483 −5.58267 1.075003 −13.1278 −5.77624 1.075003 −12.8883 −5.95435 1.075003 −12.621 −6.12168 1.075003 −12.0333 −6.39803 1.075003 −11.3321 −6.60401 1.075003 −10.5401 −6.71136 1.075003 −9.68015 −6.70462 1.075003 −8.77346 −6.57897 1.075003 −7.83446 −6.33705 1.075003 −6.87119 −5.98664 1.075003 −5.88764 −5.53401 1.075003 −4.89076 −4.98021 1.075003 −3.89552 −4.3182 1.075003 −2.92226 −3.5433 1.075003 −1.98716 −2.66272 1.075003 −1.09941 −1.69205 1.075003 −0.26324 −0.64924 1.075003 0.520834 0.447685 1.075003 1.255052 1.581436 1.075003 1.942242 2.736895 1.075003 2.584141 3.901582 1.075003 3.183908 5.063375 1.075003 3.74574 6.210687 1.075003 4.271636 7.334265 1.075003 4.76296 8.425929 1.075003 5.222064 9.477624 1.075003 5.650404 10.48245 1.075003 6.048153 11.43485 1.075003 6.416041 12.32959 1.075003 6.754675 13.16223 1.075003 7.063931 13.92944 1.075003 7.343901 14.62857 1.075003 7.59481 15.25775 1.075003 7.816825 15.81606 1.075003 8.010254 16.30349 1.075003 8.175714 16.72089 1.075003 8.314068 17.07012 1.075003 8.426533 17.35405 1.075003 8.514691 17.57657 1.075003 8.562587 17.74999 1.075003 8.53944 17.87256 1.075003 8.499157 17.93792 1.075003 8.457457 17.97679 1.075003 8.415351 18.00094 1.075003 8.355537 18.01692 1.075003 8.259625 18.00414 1.075003 8.149987 17.90874 1.075003 8.054665 17.72495 1.075003 7.937315 17.48754 1.075003 7.793241 17.19536 1.075003 7.621284 16.84597 1.075003 7.420732 16.43775 1.075003 7.191165 15.96991 1.075003 6.932473 15.44243 1.075003 6.644794 14.856 1.075003 6.328301 14.21217 1.075003 5.983447 13.51307 1.075003 5.610344 12.7618 1.075003 5.209214 11.96204 1.075003 4.781118 11.11757 1.075003 4.326746 10.233 1.075003 3.843898 9.315355 1.075003 3.326555 8.375243 1.075003 2.763346 7.428984 1.075003 2.141038 6.499456 1.075003 1.446985 5.618034 1.075003 0.675906 4.817731 1.075003 −0.17511 4.121662 1.075003 −1.10845 3.552341 1.075003 −2.11363 3.123046 1.075003 −3.1738 2.838839 1.075003 −4.26632 2.688389 1.075003 −5.36698 2.636176 1.075003 −6.45419 2.626505 1.075003 −7.50195 2.600138 1.075003 −8.47553 2.518738 1.075003 −9.34696 2.373145 1.075003 −10.1152 2.168959 1.075003 −10.8052 1.910018 1.075003 −11.4492 1.587902 1.075003 −12.0683 1.185469 1.075003 −12.6606 0.687817 1.075003 −13.2043 0.091877 1.075003 −13.5175 −0.34332 1.075003 −13.7988 −0.8259 1.075003 −14.0207 −1.31004 1.075003 −14.179 −1.76204 1.075003 −14.284 −2.17246 1.075003 −14.349 −2.55579 1.075003 −14.3813 −2.94409 1.075003 −14.9886 −5.43265 1.150007 −14.7868 −5.95502 1.150007 −14.5681 −6.36183 1.150007 −14.353 −6.67466 1.150007 −14.1385 −6.92981 1.150007 −13.918 −7.14935 1.150007 −13.692 −7.33961 1.150007 −13.4624 −7.50401 1.150007 −13.2306 −7.64572 1.150007 −12.9884 −7.7718 1.150007 −12.3653 −8.01057 1.150007 −11.6274 −8.16808 1.150007 −10.8025 −8.21465 1.150007 −9.91703 −8.13499 1.150007 −8.99555 −7.9249 1.150007 −8.05509 −7.58738 1.150007 −7.10579 −7.12924 1.150007 −6.15034 −6.56041 1.150007 −5.18959 −5.89241 1.150007 −4.23041 −5.12804 1.150007 −3.28875 −4.26465 1.150007 −2.38371 −3.3027 1.150007 −1.52806 −2.25271 1.150007 −0.72871 −1.13033 1.150007 0.014681 0.04561 1.150007 0.704732 1.257513 1.150007 1.345199 2.489687 1.150007 1.938393 3.72927 1.150007 2.487789 4.964009 1.150007 2.998445 6.18201 1.150007 3.473185 7.37365 1.150007 3.913925 8.530504 1.150007 4.323501 9.644372 1.150007 4.703995 10.70812 1.150007 5.056013 11.71595 1.150007 5.380622 12.66251 1.150007 5.678808 13.5432 1.150007 5.950673 14.35455 1.150007 6.196511 15.09381 1.150007 6.416687 15.75904 1.150007 6.611434 16.34932 1.150007 6.781075 16.86462 1.150007 6.926188 17.30588 1.150007 7.04754 17.67507 1.150007 7.146202 17.97522 1.150007 7.223796 18.2104 1.150007 7.26648 18.39182 1.150007 7.239933 18.51868 1.150007 7.196285 18.58542 1.150007 7.151907 18.62435 1.150007 7.107696 18.64765 1.150007 7.045672 18.66144 1.150007 6.948353 18.64281 1.150007 6.843257 18.53623 1.150007 6.757007 18.34082 1.150007 6.651602 18.08874 1.150007 6.522207 17.77854 1.150007 6.367771 17.40768 1.150007 6.187651 16.97443 1.150007 5.981421 16.478 1.150007 5.748996 15.91835 1.150007 5.490447 15.29623 1.150007 5.205842 14.61325 1.150007 4.895507 13.87166 1.150007 4.559394 13.07464 1.150007 4.197443 12.22605 1.150007 3.810781 11.32972 1.150007 3.400282 10.39018 1.150007 2.962481 9.414985 1.150007 2.487908 8.416319 1.150007 1.962607 7.411832 1.150007 1.366155 6.42926 1.150007 0.679489 5.506991 1.150007 −0.10413 4.68401 1.150007 −0.98802 3.985503 1.150007 −1.97513 3.435063 1.150007 −3.05342 3.046292 1.150007 −4.20088 2.817981 1.150007 −5.38852 2.719621 1.150007 −6.58564 2.699511 1.150007 −7.75822 2.69523 1.150007 −8.86152 2.648183 1.150007 −9.84344 2.52848 1.150007 −10.6754 2.34057 1.150007 −11.3774 2.099052 1.150007 −12.0051 1.801458 1.150007 −12.6142 1.419724 1.150007 −13.2289 0.913998 1.150007 −13.831 0.255033 1.150007 −14.3665 −0.54768 1.150007 −14.6707 −1.16381 1.150007 −14.9169 −1.83487 1.150007 −15.082 −2.48255 1.150007 −15.1742 −3.0589 1.150007 −15.2138 −3.55699 1.150007 −15.2171 −4.00565 1.150007 −15.1872 −4.458 1.150007 −15.1134 −4.94417 1.150007

In the above Table 1, the plane Zadim=0 corresponds to the reference plane P0 situated at the base of the profile. This plane P0 is the intersection of the stacking axis AE of the set of vanes with the axisymmetric surface of the hub, i.e. it passes via the intersection between the axis AE and the surface where it is assembled with the inner shroud. The plane Zadim=1 corresponds to the reference plane P1 situated at the top of the profile. This plane P1 is the intersection between the axis AE and the axisymmetric surface of the casing, i.e. it passes via the intersection of the axis AE with the surface where it is assembled with the outer ring.

By convention, the stacking axis AE of a vane of a nozzle, is an axis that extends in the radial direction Z and is an axis on which the various X,Y sections of the set of vanes are “stacked” when designing the set of vanes.

As mentioned at the beginning of the present description, the aerodynamic profile of the disclosure is substantially identical to the nominal profile defined in the above table, i.e. it departs from said nominal profile by very little at most, being defined in particular within an envelope of ±1 mm in a direction normal to the surface of the nominal profile, and/or having X,Y coordinates lying within a range of ±5% relative to the coordinates X,Y of the nominal profile. Also, the aerodynamic profile is positioned rotated within ±10°, particularly within ±5°, more particularly within ±3°, with respect to the stacking axis of the vane. In other words, the stagger angle may vary by ±10°, particularly by ±5°, more particularly by ±3°.

The coordinate Zadim is non-dimensional, i.e. for a point P, situated at a distance D from the plane P0 (where D is measured along the axis Z), the value of Zadim is D/H, where H represents the total height of the profile as measured between the planes P0 and P1. Naturally, in the Table 1 above, by multiplying the coordinate Zadim by the height H, it is possible to obtain the complete coordinates of the vane.

By way of example, the total height H is preferably about 70 mm ±10%. By way of example, the plane P0 is situated at a distance of about 397 mm ±10% from the axis A.

In the above table, the profile is characterized by fifteen section planes at constant Zadim coordinates, for which the coordinates X and Y are specified. These fifteen section planes include fifteen section planes located between Zadim=0 and Zadim=1, and thus arranged in the fluid stream, these section planes being distributed from P0 to P1. The fifteen section planes also include four section planes which are substantially outside the fluid stream (two section planes with Zadim<0, and two section planes with Zadim>1), these four section planes being provided for ensuring geometric continuity of the stacking, close to the head and the foot, respectively. In each section plane at constant coordinate Zadim, the section of the profile is given by a continuous and smooth curve, interconnecting all of the points (X,Y). In each section plane, the profile is interpolated so as to generate a uniform profile. Between the section planes, the profile is to be interpolated so as to generate a complete vane, as homogeneous as possible.

It is advantageous for the nozzle that includes the vane of the disclosure to have 155 to 175 vanes, more particularly 165 vanes, that present aerodynamic profiles as defined above. 

What is claimed is:
 1. An aerodynamic profile for a turbine vane, the profile being, when cold and in a non-coated state, substantially identical to a nominal profile determined by the Cartesian coordinates X,Y,Zadim given in Table 1, in which the coordinate Zadim is the quotient D/H, where D is the distance of the point under consideration from a reference X,Y plane situated at the base of the nominal profile, and H is the height of said profile measured from said reference plane out to the end of the vane, the measurements D and H being taken radially relative to the axis of the turbine, while the coordinate X is measured in the axial direction of the turbine.
 2. The aerodynamic profile as claimed in claim 1, wherein said profile is defined within an envelope of ±1 mm in a direction normal to the surface of the nominal profile.
 3. The aerodynamic profile as claimed in claim 1, wherein the coordinates X,Y of said profile lie within a range of ±5% relative to the coordinates X,Y of the nominal profile.
 4. The aerodynamic profile as claimed in claim 1, wherein the aerodynamic profile is positioned rotated within ±10° with respect to the radial axis of the vane, along which the coordinate Zadim is measured.
 5. The aerodynamic profile as claimed in claim 1, wherein the aerodynamic profile is positioned rotated within ±5° with respect to the radial axis of the vane, along which the coordinate Zadim is measured.
 6. The aerodynamic profile as claimed in claim 1, wherein the aerodynamic profile is positioned rotated within ±3° with respect to the radial axis of the vane, along which the coordinate Zadim is measured.
 7. The aerodynamic profile as claimed in claim 1, wherein the vane is a nozzle vane forming a part of a stator of a turbine.
 8. A turbine vane, presenting the aerodynamic profile as claimed in claim
 1. 9. A turbine, including turbine vanes presenting aerodynamic profiles as claimed in claim
 1. 10. A turbine as claimed in claim 9, including a nozzle that is stationary in rotation, having 155 to 175 vanes that present the aerodynamic profiles.
 11. A turbine as claimed in claim 9, including a nozzle that is stationary in rotation, having 165 vanes that present the aerodynamic profiles.
 12. A turbine nozzle forming a portion of a turbine stator, wherein all the vanes of the nozzle present the aerodynamic profile as claimed in claim
 1. 13. A turbine nozzle as claimed in claim 12, including 155 to 175 vanes that present the aerodynamic profiles.
 14. A turbine nozzle as claimed in claim 12, including 165 vanes that present the aerodynamic profiles. 